Question

Make up a radical equation that has no solution.

Short answer

The equation \( \sqrt{x+1} = -x \) has no solution because the square root is always non-negative while the right-hand side can be negative.

Step-by-step solution

- Understand Radical Equations

A radical equation is an equation in which a variable is inside a radical, usually a square root. The general form of a radical equation is \( \sqrt{f(x)} = g(x) \).

- Create Example Equation

To create a radical equation that has no solution, we need to ensure the contents under the radical cannot equal the other side of the equation.\ For example, consider the equation \( \sqrt{x+1} = -x \).

- Analyze the Equation

Recall that the square root of a number is always non-negative, so \( \sqrt{x+1} \geq 0 \). However, the right-hand side of our equation is \( -x \), which can be negative or positive depending on the value of \( x \).

- Identify No Solution

For the equation \( \sqrt{x+1} = -x \) to have any solution, both sides of the equation must be non-negative. But since \( \sqrt{x+1} \geq 0 \) and \(-x \leq 0\), they can never be equal for any real value of \( x \). Therefore, the equation has no solution.

Key concepts

square root

The square root is a fundamental concept in mathematics. It deals with finding a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 * 3 = 9.
When working with square roots, especially in equations, it is important to remember that the square root of a non-negative number is also non-negative. This means if we have a square root, like \(\sqrt{a}\), it can never be negative as long as 'a' is a non-negative number. This property plays a critical role in solving radical equations.

no solution

An equation has no solution when no value for the variable can satisfy it. In the case of radical equations, this can often happen due to the properties of square roots.
For example, consider the equation \(\sqrt{x+1} = -x\). The left side is a square root and hence always non-negative. But the right side is \(-x\) which can be negative if 'x' is positive.
Since a non-negative number cannot equal a negative number, there is no real number 'x' that can satisfy this equation, so it has no solution.

variable under radical

In a radical equation, the variable is often inside a radical, such as within a square root. This makes the equation different and sometimes more challenging to solve compared to regular equations.
When solving radical equations, it is crucial to isolate the radical expression first and then square both sides of the equation.
However, care must be taken since squaring both sides can introduce extraneous solutions. Always check your solutions in the original equation. In our example, \(\sqrt{x+1} = -x\), even after isolating the radical and squaring both sides, we still find that no real 'x' satisfies this equation due to the properties of square roots and negative numbers.